Up: The Platonic Solids

How are the Platonic Solids Made?

should there be pictures?

What to consider

If we want to construct regular polyhedra, we have to make a few choices. Since there can be only one shape for the faces of the polyhedron, we must choose the shape. We also have to make sure that there are the same number of faces (at least three) around each vertex.

Getting started with triangular faces

We will start with the simplest regular polygon: the equilateral triangle. Joining the edges of three equilateral triangles gives a prism with a triangular base. In fact, the base is also an equilateral triangle. Attaching an equilateral triangle to the base yields the tetrahedron, the regular solid with three triangles meeting at every vertex.
 

We have just seen that we can build a regular polyhedron using equilateral triangles, arranged with three at every vertex. Can we arrange them so that four triangles meet at every vertex? Yes. Joining four equilateral triangles along their edges gives a prism with a square base. Making two such prisms and joining their bases together creates an octahedron, the regular solid with four triangles meeting at every vertex.
 

Ok, now how about putting five equilateral triangles around a vertex? Again, arranging five triangles around a vertex yields a prism with a pentagonal base. We can't simply join two such prisms, as we did in constructing the octahedron, because some vertices will only have four faces around them. We have to make a ring of ten triangles, arranged so that they alternate pointing-up and pointing-down. Attatching the two prisms at the top and bottom of the ring of triangles completes the icosahedron, the regular solid with five triangles meeting at every vertex.
 

If we arrange six equilateral triangles around a vertex, they will lie flat in the plane, so there is nothing more we can do with triangles. You can see a picture of six equilateral triangles around a vertex.



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